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The derivative of a polynomial $p(z)$ of degree $n$, with respect to point $\alpha$ is defined by $D_{\alpha}p(z)=np(z)+(\alpha-z)p'(z)$. Let $p(z)$ be a polynomial having all its zeros in the unit disk $|z| \leq 1$. The Sendov conjecture asserts that if all the zeros of a polynomial $p(z)$ lie in the closed unit disk, then there must be a zero of $p'(z)$ within unit distance of each zero. In this paper, we obtain certain results concerning the location of the zeros of $D_{\alpha}p(z)$ with respect to a specific zero of $p(z)$ and a stronger result than Sendov conjecture is obtained. Further, a result is obtained for zeros of higher derivatives of polynomials having multiple roots.
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참고문헌 (7)
참고문헌 신청
A. Aziz / 1984 / On the location of critical points of Polynomials / J. Austral. Math. Soc 36:4~11
B. Bojanov / 1985 / On a conjecture of Sendov about critical points of a polynomial Mathematische Zeitschrift 190:281~286
J. E. Brown / 1999 / Proof of Sendov conjecture for polynomials of degree at most eight / J. Math. Anal. Appl 232:272~292
A. W. Goodman / 1969 / On the zeros of polynomial and its derivative / Proc. Amer. Math. Soc 21:273~274
Q. I Rahman / 2002 / Analytic Theory of Polynomials / Oxford University Press
B. Bojanov / 1985 / On a conjecture of Sendov about critical points of a polynomial Mathematische Zeitschrift 190:281~286
J. E. Brown / 1999 / Proof of Sendov conjecture for polynomials of degree at most eight / J. Math. Anal. Appl 232:272~292
A. W. Goodman / 1969 / On the zeros of polynomial and its derivative / Proc. Amer. Math. Soc 21:273~274
Q. I Rahman / 2002 / Analytic Theory of Polynomials / Oxford University Press
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